The number of equivalence relations on the set $\{1,2,3,4,5\}$ is

since each equivalence relation on a set yields a partition of that set in disjoint equivalence classes, for a finite set, the number of equivalence relations is the number of partitions.

if B_n is the number of partitions on a set of size n, notice $B_{n+1}=\sum _{k=0}^{n}C\left(n,k\right)B_k\text{, and }{B}_0=1$

$\begin{array}{rcl}{B}_4& =& \sum _{k=0}^{3}C\left(3,k\right)B_k\\ & =& C\left(3,0\right)B_0+C\left(3,1\right)B_1+C\left(3,2\right)B_2+C\left(3,3\right)B_3\\ & =& 1+3\left(1\right)+3\left(2\right)+5\\ & =& 15\end{array}$

$\begin{array}{rcl}{B}_5& =& \sum _{k=0}^{4}C\left(4,k\right)B_k\\ & =& C\left(4,0\right)B_0+C\left(4,1\right)B_1+C\left(4,2\right)B_2+C\left(4,3\right)B_3+C\left(4,4\right)B_4\\ & =& 1\left(1\right)+4\left(1\right)+6\left(2\right)+4\left(5\right)+B_4\\ & =& 37+15\\ & =& 52\end{array}$

$B_0=1,B_1=1,B_2=2,B_3=5,B_4=15,B_5=52.$

$\Rightarrow \text{ The number of equivalence relations on a five element set is }52\text{. }$